Spectral Measures for Derivative Powers via Matrix-Valued Clark Theory

TL;DR

This paper develops a matrix-valued Clark theory approach to analyze spectral measures of derivative operator powers, providing explicit formulas and a comprehensive spectral characterization beyond traditional differential equation methods.

Contribution

It introduces a novel application of finite-rank perturbations and matrix-valued analytic functions to compute spectral measures for derivative powers, expanding spectral analysis tools.

Findings

Explicit formulas for spectral measures of derivative powers.

Support and weights of Clark measures are characterized.

Provides a complete spectral description beyond eigenvalues.

Abstract

The theory of finite-rank perturbations allows for the determination of spectral information for broad classes of operators using the tools of analytic function theory. In this work, finite-rank perturbations are applied to powers of the derivative operator, providing a full account from self-adjoint boundary conditions to computing aspects of the operators' matrix-valued spectral measures. In particular, the support and weights of the Clark (spectral) measures are computed via the connection between matrix-valued contractive analytic functions and matrix-valued nonnegative measures through the Herglotz Representation Theorem. For operators associated with several powers of the derivative, explicit formulae for these measures are included. While eigenfunctions and eigenvalues for these operators with fixed boundary conditions can often be computed using direct methods from ordinary differential equations, this approach provides a more complete picture of the spectral information.